Approximating Probability using Monte Carlo Method

I have a problem where I need to sample two random points (x and y) unifromrly from the unit square [0,1] x [0,1]

I need to use the Monte Carlo Method to approximate the probability that (the 2 norm of x-y) ||x-y||_2 is smaller or equal to 1/2

Anyone has an idea on how to go on about this?

[James Tursa]

What have you tried so far? What specific problems are you having with your code?

[John D'Errico]

An interesting question is how to compute that result analytically, without recourse to a simulation. Hmm...

'Til then so you can verify your result, I get with a sample size of 1e8 points:

p =

0.48335794

With a predicted standard deviation on that estimate of

s =

4.99722965090615e-05

[khalid alharthi]

This is what I have, the thing I am not sure about is how to use the Monte Carlo Method to obtain a probability.

for i=1:1000
   numPoints = 1000000;
x = rand(numPoints, 1);
y = rand(numPoints, 1);

f = norm(x);
g = norm(y);

Z(i) = abs(f-g);
    
end

R = Z <= 1/2;
U = R(R==1);
Probability = length(U)/length(R)

Answer 1

[Image Analyst]

Hint:

xy = rand(1000000, 2);

or

numPoints = 1000000;
x = rand(numPoints, 1);
y = rand(numPoints, 1);

Look up norm in the help.